# On a problem of Bharanedhar and Ponnusamy involving planar harmonic mappings

## TL;DR

This paper disproves a previous conjecture about the univalence of certain harmonic mappings and introduces new results on a subclass of close-to-convex harmonic functions.

## Contribution

It provides a negative answer to a longstanding problem on harmonic univalence and explores properties of a new subclass of close-to-convex harmonic mappings.

## Key findings

- The class contains non-univalent functions for all parameter values.
- New results on a subclass of close-to-convex harmonic mappings.
- Disproof of a previous univalence conjecture.

## Abstract

In this paper, we give a negative answer to a problem presented by Bharanedhar and Ponnusamy (Rocky Mountain J. Math. 44: 753--777, 2014) concerning univalency of a class of harmonic mappings. More precisely, we show that for all values of the involved parameter, this class contains a non-univalent function. Moreover, several results on a new subclass of close-to-convex harmonic mappings, which is motivated by work of Ponnusamy and Sairam Kaliraj (Mediterr. J. Math. 12: 647--665, 2015), are obtained.